It turns out I was wrong in my logic. What ended up happening was Roma (3rd) took Napoli's (2nd) group spot, and Napoli took Roma's playoff spot. It seems winning the UEFA Cup just means you bump whoever has the last group spot in your league. I always thought the UEFA winner spot was factored in as its own qualification spot and whichever lucky team (me in this case) finished behind the UEFA Cup winner in their domestic league would inherit their qualification spot.
Oh and Juventus finished in 1st and won the Champions League. Wait so...
This is a bit messy of an answer, so feel free to not read it.
The Europa League winner goes to the non-champion playoff at least, but can make the group stage straight away. Roma would only go straight to the group stages if the winner of the Champions League already qualified for the group stages through their respective league. From UEFA: "The Europa League champion may be promoted into the group stage if the Champions League champion qualifies for the group stage through their domestic competition." Since Juventus won Serie A and the Champions League, they use their winner of Italy spot for the group stages and Roma gets bumped up to the group.
Italy, being ranked 4th only has 3 spots. It may get four if the winner of the Champions League OR the Europa League finishes outside of the normal three allocations within Italy. It's the same for every country with the exception of leagues ranked 13th and lower with a double exception for Liechtenstein (or Luxembourg, one of those rich L central european countries).
With Juve winning the Champs League and coming first, Roma got bumped into the group stages at the expense of Napoli because both winners were in the top three. So Napoli goes to the Non-champion playoffs.
The really messy part comes in if the Champions League winner did not qualify for the group stages via their league position. Think Chelsea a few years back. Under the new rules there are only 10 teams who make it to the Non-champion playoff round, but 11 spots:
5 winners from the third qualifying round for non-champions;
2 third-placed teams from associations 4–5 (Italy and Portugal)*;
3 fourth-placed teams from associations 1–3 (Spain, England, Germany)*; and
the Europa League winners.
The Europa League spot may kick out one of the other 10 spots, how depends on where they finished and what league they're in.
One scenario is that the Roma won the Europa League and finished third and the winner of the Champs League finished outside of their league qualifying spots (for the sake of the argument, I'm going to use Sassuolo as the example so let's pretend they made the Champs League from last season and somehow won it). Sassuolo would be in the group stages along with Juve and Napoli and Roma would stay in the non-champion playoff because their Europa League spot would kick out their 3rd place in Italy spot.
So Sassuolo won the Champs League, Roma won the Europa League, both outside of top 3 and Milan finished third. Sassuolo, Juve, Napoli are in the group stages. Roma in the non-champion playoffs, Milan in
the non-champion 3rd round quali along with the 3rd place team from Portugal.
And just so I have another scenario covered. If Juve won the Champs League, Roma won the Europa League. Top 4 in order are Juve, Napoli, Milan, Roma. Juve, Napoli, and Roma make the group stages, Milan makes the non-champions playoff.
* based on coefficients and seedings for 2016/2017 season.
Edit:Had the incorrect outcome for the second to last example, fixed it in bold.